If the domain of the function f(x) = cos^−1( 2−|x| 4 ) + (loge(3−x))^−1 is [−α, β) − {y}, then α + β + γ is equal to :

Question

If the domain of the function \(f(x)=\cos ^{-1}\left(\frac{2-|x|}{4}\right)+\left(\log _{e}(3-x)\right)^{-1}\) is \([-\alpha, \beta)-\{y\}\), then \(\alpha+\beta+\gamma\) is equal to :

Answer 1

\(-1 \leq\left|\frac{2-|x|}{4}\right| \leq 1\)

\(\Rightarrow\left|\frac{2-|x|}{4}\right| \leq 1\)

\(-4 \leq 2-|x| \leq 4\)

\(-6 \leq-|x| \leq 2\)

\(-2 \leq|x| \leq 6\)

\(|x| \leq 6\)

\(\Rightarrow x \in[-6,6]\quad ....(1)\)

Now, \(3-x \neq 1\)

And \(x \neq 2 \quad ....(2)\)

and \(3-x>0\)

\(x<3\quad....(3)\)

From (1), (2) and (3)

\(\Rightarrow x \in[-6,3)-\{2\}\)

\(\alpha=6\)

\(\beta=3\)

\(\gamma=2\)

\(\alpha+\beta+\gamma=11\)